![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dvcosre | Structured version Visualization version GIF version |
Description: The real derivative of the cosine. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
dvcosre | ⊢ (ℝ D (𝑥 ∈ ℝ ↦ (cos‘𝑥))) = (𝑥 ∈ ℝ ↦ -(sin‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reelprrecn 11208 | . . 3 ⊢ ℝ ∈ {ℝ, ℂ} | |
2 | cosf 16075 | . . 3 ⊢ cos:ℂ⟶ℂ | |
3 | ssid 4004 | . . 3 ⊢ ℂ ⊆ ℂ | |
4 | nfcv 2902 | . . . . . 6 ⊢ Ⅎ𝑥ℝ | |
5 | nfrab1 3450 | . . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∈ ℂ ∣ -(sin‘𝑥) ∈ V} | |
6 | 4, 5 | dfss2f 3972 | . . . . 5 ⊢ (ℝ ⊆ {𝑥 ∈ ℂ ∣ -(sin‘𝑥) ∈ V} ↔ ∀𝑥(𝑥 ∈ ℝ → 𝑥 ∈ {𝑥 ∈ ℂ ∣ -(sin‘𝑥) ∈ V})) |
7 | recn 11206 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
8 | 7 | sincld 16080 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (sin‘𝑥) ∈ ℂ) |
9 | 8 | negcld 11565 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → -(sin‘𝑥) ∈ ℂ) |
10 | elex 3492 | . . . . . . 7 ⊢ (-(sin‘𝑥) ∈ ℂ → -(sin‘𝑥) ∈ V) | |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → -(sin‘𝑥) ∈ V) |
12 | rabid 3451 | . . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ ℂ ∣ -(sin‘𝑥) ∈ V} ↔ (𝑥 ∈ ℂ ∧ -(sin‘𝑥) ∈ V)) | |
13 | 7, 11, 12 | sylanbrc 582 | . . . . 5 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ {𝑥 ∈ ℂ ∣ -(sin‘𝑥) ∈ V}) |
14 | 6, 13 | mpgbir 1800 | . . . 4 ⊢ ℝ ⊆ {𝑥 ∈ ℂ ∣ -(sin‘𝑥) ∈ V} |
15 | dvcos 25836 | . . . . 5 ⊢ (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)) | |
16 | 15 | dmmpt 6239 | . . . 4 ⊢ dom (ℂ D cos) = {𝑥 ∈ ℂ ∣ -(sin‘𝑥) ∈ V} |
17 | 14, 16 | sseqtrri 4019 | . . 3 ⊢ ℝ ⊆ dom (ℂ D cos) |
18 | dvres3 25763 | . . 3 ⊢ (((ℝ ∈ {ℝ, ℂ} ∧ cos:ℂ⟶ℂ) ∧ (ℂ ⊆ ℂ ∧ ℝ ⊆ dom (ℂ D cos))) → (ℝ D (cos ↾ ℝ)) = ((ℂ D cos) ↾ ℝ)) | |
19 | 1, 2, 3, 17, 18 | mp4an 690 | . 2 ⊢ (ℝ D (cos ↾ ℝ)) = ((ℂ D cos) ↾ ℝ) |
20 | ffn 6717 | . . . . . . 7 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ) | |
21 | 2, 20 | ax-mp 5 | . . . . . 6 ⊢ cos Fn ℂ |
22 | dffn5 6950 | . . . . . 6 ⊢ (cos Fn ℂ ↔ cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥))) | |
23 | 21, 22 | mpbi 229 | . . . . 5 ⊢ cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)) |
24 | 23 | reseq1i 5977 | . . . 4 ⊢ (cos ↾ ℝ) = ((𝑥 ∈ ℂ ↦ (cos‘𝑥)) ↾ ℝ) |
25 | ax-resscn 11173 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
26 | resmpt 6037 | . . . . 5 ⊢ (ℝ ⊆ ℂ → ((𝑥 ∈ ℂ ↦ (cos‘𝑥)) ↾ ℝ) = (𝑥 ∈ ℝ ↦ (cos‘𝑥))) | |
27 | 25, 26 | ax-mp 5 | . . . 4 ⊢ ((𝑥 ∈ ℂ ↦ (cos‘𝑥)) ↾ ℝ) = (𝑥 ∈ ℝ ↦ (cos‘𝑥)) |
28 | 24, 27 | eqtri 2759 | . . 3 ⊢ (cos ↾ ℝ) = (𝑥 ∈ ℝ ↦ (cos‘𝑥)) |
29 | 28 | oveq2i 7423 | . 2 ⊢ (ℝ D (cos ↾ ℝ)) = (ℝ D (𝑥 ∈ ℝ ↦ (cos‘𝑥))) |
30 | 15 | reseq1i 5977 | . . 3 ⊢ ((ℂ D cos) ↾ ℝ) = ((𝑥 ∈ ℂ ↦ -(sin‘𝑥)) ↾ ℝ) |
31 | resmpt 6037 | . . . 4 ⊢ (ℝ ⊆ ℂ → ((𝑥 ∈ ℂ ↦ -(sin‘𝑥)) ↾ ℝ) = (𝑥 ∈ ℝ ↦ -(sin‘𝑥))) | |
32 | 25, 31 | ax-mp 5 | . . 3 ⊢ ((𝑥 ∈ ℂ ↦ -(sin‘𝑥)) ↾ ℝ) = (𝑥 ∈ ℝ ↦ -(sin‘𝑥)) |
33 | 30, 32 | eqtri 2759 | . 2 ⊢ ((ℂ D cos) ↾ ℝ) = (𝑥 ∈ ℝ ↦ -(sin‘𝑥)) |
34 | 19, 29, 33 | 3eqtr3i 2767 | 1 ⊢ (ℝ D (𝑥 ∈ ℝ ↦ (cos‘𝑥))) = (𝑥 ∈ ℝ ↦ -(sin‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 {crab 3431 Vcvv 3473 ⊆ wss 3948 {cpr 4630 ↦ cmpt 5231 dom cdm 5676 ↾ cres 5678 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 ℝcr 11115 -cneg 11452 sincsin 16014 cosccos 16015 D cdv 25713 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-fi 9412 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ico 13337 df-icc 13338 df-fz 13492 df-fzo 13635 df-fl 13764 df-seq 13974 df-exp 14035 df-fac 14241 df-bc 14270 df-hash 14298 df-shft 15021 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15640 df-ef 16018 df-sin 16020 df-cos 16021 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-mulg 18994 df-cntz 19229 df-cmn 19698 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-fbas 21231 df-fg 21232 df-cnfld 21235 df-top 22717 df-topon 22734 df-topsp 22756 df-bases 22770 df-cld 22844 df-ntr 22845 df-cls 22846 df-nei 22923 df-lp 22961 df-perf 22962 df-cn 23052 df-cnp 23053 df-haus 23140 df-tx 23387 df-hmeo 23580 df-fil 23671 df-fm 23763 df-flim 23764 df-flf 23765 df-xms 24147 df-ms 24148 df-tms 24149 df-cncf 24719 df-limc 25716 df-dv 25717 |
This theorem is referenced by: itgsin0pilem1 45128 itgsinexplem1 45132 fourierdlem39 45324 |
Copyright terms: Public domain | W3C validator |